Infinitesimal Calculus of Random Walk and Poisson Processes

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Gauge Institute Journal, H. Vic Dannon 12. t= b ∫ t= a ftdB () (, ζ t) While EB [ ( ζ , t)] has unbounded Variation in [,] ab, integration with respect to B(,) ζ t is possible. Let f () t be a hyper-real function on the bounded time interval [,] ab . f () t need not be bounded. At each a ≤ t ≤ b, there is a Bernoulli Random Variable dB(,) ζ t = B(, ζ t + dt) − B(,) ζ t = B (,) ζ t = B (,) ζ t dt. We form the Integration Sum For any dt , t= b t= b t= b ∑ ∑ ∑ f () tdB(, ζ t) = ftB () (, ζ t) = ftB () (, ζ tdt ) i t= a t= a t= a (1) the First Moment of the Integration Sum is ⎡t= b ⎤ t= b E f() t B(, ζ t) dt = f() t E[ B ∑ ∑ (, ζ t)] dt = 0. ⎢⎣t= a ⎥⎦ t= a (2) the Second Moment of the Integration sum is i 0 48
Gauge Institute Journal, H. Vic Dannon E ⎡ ⎛ ⎞ 2 t= b ⎤ t= b τ= b ⎢⎜ f () t B (, ) i t ⎡⎛ ⎞⎛ E f () t B i(, t ⎞⎤ ζ ζ ) f ( τ) B (, ζ τ) ∑ = ∑ ∑ j ⎜ ⎝t= a ⎠⎟ ⎢⎝⎜t= a ⎠⎝ ⎟⎜ ⎣ τ= a ⎠⎟ ⎢ ⎥ ⎥ ⎣ ⎦ ⎦ t= b τ= b = ∑∑ t= a τ= a f ()( tfτ) EB [ (, ζ τ) B(, ζ t)] Since the Bernoulli Random Variables are independent, EB [ ( ζτ , ) B( ζ, t)] = EB [ ( ζ, t)] = ( dx) j 2 2 j i i i only for t = τ . Then, ⎡ ⎤ ⎛ E ⎜ f t B t f t ⎢ ⎜⎝ ⎣ ⎥⎦ 2 2 t= b ⎞ t= b 2 2 ∑ () i(, ζ ) ⎟ = ∑ ()( dx) , t= a ⎠⎟ ⎥ t= a (2 Ddt ) t= b 2 = 2 D∑ f ( t) dt, t= a t= b 2 = 2 D∫ f ( t) dt. t= a assuming (d x) = (2 D) dt , and f () t integrable Thus, for any dt , the Integration Sum is a unique well- defined hyper-real Random Variable I() ζ . We call I() ζ the integral of f () t , with respect to B(, ζ t) from x t= b = a, to x = b, and denote it by f () tdB(, ζ t) . ∫ t= a 49
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Infinitesimal Calculus of Random Walk and Poisson Processes

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